What question did this study set out to answer?

The research aims to establish a framework for closed-form expressions of the complex-order Riemann zeta function and its generalizations.

February 2, 2026Open Access

Closed-Form Expressions and Systematic Theory for Complex-Order Riemann Zeta Function and Its Extensions

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Develops closed-form expressions and systematic theory for complex-order Riemann zeta functions, indicating broad applications.

The research aims to establish a framework for closed-form expressions of the complex-order Riemann zeta function and its generalizations.
Employed theory of complex functions and holomorphic continuation of the Gamma function.
Utilized contour integration to extend real-order integrals to the complex plane.
Developed representations involving complex-order hypergeometric functions.
Investigated complex-order fractional differential equations and their structures.
Conducted numerical verification through interval arithmetic.
Provided a universal closed-form expression applicable to complex numbers.
Explored properties of special values for complex algebraic inputs.
Created a Γ-rational approximation theory with error estimates in complex regions.
Developed algorithms for precise contour integration and hypergeometric series.
Extended the theory to multiple zeta functions and complex polylogarithms.

shifa liu (2025) studied this question.

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